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I have two Likert-scale variables, that values vary from "strongly disagree" to "strongly agree". Somehow I know, that many people who stronly disagree on the first scale, strongly agree on the second one (and the other way round). People who are moderate on one scale, they are also moderate on the other scale.

So my question is: is there any way to measure a curvilinear relationship between ordinal variables?

My data set is:

  matr = matrix(c(19,21,35,19,13,
                  16,28,60,24,12,
                  27,39,54,53,32,
                  43,52,32,46,48,
                  54,48,29,45,50),
              5,5)

Kendall's tau coef. is: 0.0097, which misleadingly indicates no relationship.

[UPDATE]: I'm aware that "curvilinear" in these Likert-scales case is a big word, but I see the pattern in the form "U", when I plot counts. The shoulders of "U" are those people, who "stronly disagree"/"strongly agree" with statement A and "stronly agree" with statement B. Others are moderate in both statements (the lower part of "U").

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    $\begingroup$ Kendall's tau should be exactly the thing for this situation, and one would expect a negative sign for it. I may be misunderstanding you. Could you clarify what you mean by "curvilinear relationship"? What you describe is a simple negative correlation between ordinal variables, nothing curvilinear about it. $\endgroup$ – Stephan Kolassa Nov 11 '14 at 16:29
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There's a distinctly non-monotonic association between the two sets of answers, so Kendall's tau isn't any help in measuring it.

enter image description here

The "spread" of B (rather than its mean) is related to A, which is why the second curve (which measures the mean) barely moves. (Those red lines are lowess smooths, rather than raw means by category.)

One interesting approach to investigating this relation among ordered variables is to consider a glm for the counts in terms of orthogonal polynomials. Since you mentioned quadratic relationships, it looks like a quadratic*quadratic model captures a fair amount of the variation, and indeed looking at it, most of that is captured by a quadratic (in rows) * linear in columns model.

Most of the usual ordinal-ordinal association statistics measure monotonicity. One approach if you're looking for some general measure of non-monotonic association, would be to look at nominal-nominal (such as Cramer's V), and nominal-ordinal association (e.g. see ${[\text{1}]}$).

There are many other ways to measure association between variables like these.

The first thing to do is to be clear about what you want to do with this - what it's for - it will help guide you to making choices more relevant to your problem.

[1]: Alan Agresti (1981),
"Measures of Nominal-Ordinal Association,"
Journal of the American Statistical Association, 76:375 (Sep.), pp. 524-529

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